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Comparative Analysis of Optimal Control Strategies: LQR, PID, and Sliding Mode Control for DC Motor Position Performance

Year 2023, Volume: 10 Issue: 4, 571 - 592, 31.12.2023

Hakan Kızmaz

https://doi.org/10.54287/gujsa.1393092

Abstract

This study applies these control methods to the DC motor system to examine the robustness and performance of four optimal control methods. Optimal controllers aim to control the system to minimize a selected performance index. These control methods offer advantages such as improving energy efficiency, reducing costs, and enhancing system security. The Linear Quadratic Regulator (LQR) based controller is the primary optimal control method. Two well-known traditional control techniques include the Proportional-Integral-Derivative (PID) and Integral Sliding Mode Controller (ISMC). However, they do not usually contain optimal properties. In this study, the optimal control algorithms, defined by obtaining controller parameters through the Riccati equation, are applied to achieve accurate position-tracking control in a DC motor system using Matlab/Simulink. The integral term-based algorithms seem to be robust and eliminate steady-state errors. The optimal PID controller could not provide the minimum performance index, unlike the other controllers in the study. LQR and optimal ISMC algorithms could allow the performance index to be a minimum. An illustrative comparison of the performances of all optimal control algorithms has been presented through graphical representation, along with corresponding interpretations.

Keywords

DC Motor Linear Quadratic Regulator Optimal Control Optimal PID Optimal integral sliding mode control

References

  • Anderson, B. D., & Moore, J. B. (2007). Optimal Control Linear Quadratic Methods (91.12 edition ed.). Dover Publications.
  • Aravind, M. A., Saikumar, N., & Dinesh, N. S. (2017, May 19-21). Optimal position control of a DC motor using LQG with EKF. In: Proceedings of the International Conference on Mechanical, System and Control Engineering (ICMSC), (pp. 149-154). St. Petersburg. https://www.doi.org/10.1109/ICMSC.2017.7959461
  • Burns, R. S. (2001). Advanced Control Engineering. Woburn, England: Butterworth-Heinemann.
  • Davis, J. H. (2002). Luenberger Observers. In: Foundations of Deterministic and Stochastic Control (pp. 245-254). Boston: Birkhäuser. https://www.doi.org/10.1007/978-1-4612-0071-0_8
  • Dorf, R. C., & Bishop, R. H. (2010). Modern Control Systems (12th ed.). (M. J. Horton, A. Gilfillan, A. Dworkin, S. Disanno, G. Dulles, & D. Sandin, Eds.) New Jersey, U.S.A.: Pearson.
  • Dreyfus, S. (1962). Variational problems with inequality constraints. Journal of Mathematical Analysis and Applications, 4(2), 297-308. https://doi.org/10.1016/0022-247X(62)90056-2
  • Durdu, A., & Dursun, E. H. (2019). Sliding Mode Control for Position Tracking of Servo System with a Variable Loaded DC Motor. Elektronika Ir Elekctrotechnika, 25(4), 8-16. https://www.doi.org/10.5755/j01.eie.25.4.23964
  • Edwards, C., & Spurgeon, S. K. (1998). Sliding Mode Control Theory and Applications. Boca Raton: CRC Press.
  • Eli, S. C., Idoniboyeobu, D. C., & Braide, S. L. (2023). Performance Evaluation of D.C. Motor Speed Using Sliding Mode Controller (SMC). Journal of Emerging Trends in Electrical Engineering, 5(3), 1-6.
  • Feng, X., Liu, S., Yuan, Q., Xiao, J., & Zhao, D. (2023). Research on wheel-legged robot based on LQR and ADRC. Scientific Reports, 13(15122). https://www.doi.org/10.1038/s41598-023-41462-1
  • Franklin, G. F., Powell, J. D., & Emami-Naeini, A. (2009). Feedback Control of Dynamic Systems (6th ed.). New Jersey, U.S.A.: Pearson Education.
  • Gao, W., & Hung, J. C. (1993). Variable Structure Control of Nonlinear Systems A New Approach. IEEE Transactions On Industrial Electronics, 40(1), 45-55.
  • Golub, G. H., & Loan, C. F. (2013). Matrix Computations. Baltimore: The Johns Hopkins University Press.
  • Gorczyca, P., Hajduk, K., & Kołek, K. (2011). Optimal Control of A Laboratory DC Servo Motor. Pomiary automatyka Robotyka, 15(12), 129-134.
  • He, S., Duan, X., Qu, X., & Xiao, J. (2023). Kinematic modeling and motion control of a parallel robotic antenna pedestal. Robotica, 41(11), 3275-3295. https://www.doi.org/10.1017/S0263574723000917
  • Jouini, M., Dhahri, S., & Sellami, A. (2019). Combination of integral sliding mode control design with optimal feedback control for nonlinear uncertain systems. Transactions of the Institute of Measurement and Control, 41(5), 1331-1339. https://www.doi.org/10.1177/0142331218777562
  • Khomenko, M., Voytenko, V., & Vagapov, Y. (2013). Neural Network based Optimal Control of Dc Motor Positioning System. International Journal of Automation and Control, 7(1/2), 83-104. https://www.doi.org/10.1504/IJAAC.2013.055097
  • Kirk, D. E. (1998). Optimal Control Theory: An Introduction. New York, Mineola, USA: Dover Publications, Inc.
  • Krause, P. C., Wasynczuk, O., & Sudhoff, S. D. (2002). Analysis of Electric Machinery and Drive Systems (2nd ed.). U.S.A.: John Wiley & Sons.
  • Krishnan, R. (2001). Electric Motor Drives Modeling Analysis and Control (1st ed.). New Jersey: Prentice Hall.
  • Maghfiroh, H., Anwar, M., Anwar, M., & Ma'arif, A. (2022). Improved LQR Control Using PSO Optimization and Kalman Filter Estimator. IEEE Access, 10, 18330-18337. https://www.doi.org/10.1109/ACCESS.2022.3149951
  • Mamta, & Singh, B. (2020). Optimal Control of DC motor using Equilibrium Optimization Algorithm. International Journal of Engineering Research & Technology (IJERT), 9(5), 1272-1275.
  • Mondal, R., & Dey, J. (2020). Performance Analysis and Implementation of Fractional Order 2-DOF Control on Cart-Inverted Pendulum System. IEEE Transactions On Industry Applications, 56(6), 7055-7066.
  • Mukhopadhyay, S. (1978). P.I.D. Equivalent of Optimal Regulator. Elektronics Letters, 14(25), 821-822.
  • Naidu, D. S. (2002). Optimal Control Systems (1st ed.). CRC Press.
  • Nise, N. S. (2011). Control System Engineering (6th ed.). Pomona, U.S.A.: John Wiley & Sons, Inc.
  • Ogata, K. (2010). Modern Control Engineering (5th ed.). Natick: Pearson.
  • Paraskevopoulos, P. N. (2002). Modern Control Engineering. CRC Press.
  • Pontryagin, L. S. (1986). The Mathematical Theory of Optimal Processes (Vol. Vol 4). Switzerland: Gordon and Breach Science Publishers.
  • Pratama, G. N., Setiawan, N., Saputra, S. A., Pambudi, L., Umam, A. D., & Hermansah, M. N. (2022). Optimal Quadratic Regulator PID for Motor DC. Journal of Physics: Conference Series, 2406(012002), 1-7. https://www.doi.org/10.1088/1742-6596/2406/1/012002
  • Rasheed, L. T. (2020). Optimal Tuning of Linear Quadratic Regulator Controller Using Ant Colony Optimization Algorithm for Position Control of a Permanent Magnet Dc Motor. Iraqi Journal of Computers, Communications, Control & Systems Engineering (IJCCCE), 20(3), 29-41. https://www.doi.org/10.33103/uot.ijccce.20.3.3
  • Ruderman, M., Krettek, J., Hoffmann, F., & Bertram, T. (2008). Optimal State Space Control of DC Motor. IFAC Proceedings Volumes, 41(2), 5796-5801. https://www.doi.org/10.3182/20080706-5-KR-1001.00977
  • Saputra, D. D., Ma'arif, A., Maghfiroh, H., Baballe, M. A., Tusset, A. M., Sharkawy, A.-N., & Majdoubi, R. (2023). Performance Evaluation of Sliding Mode Control (SMC) for DC Motor Speed Control. Jurnal Ilmiah Teknik Elektro Komputer dan Informatika (JITEKI), 9(2), 502-510.
  • Simon, D. (2006). Optimal State Estimation. Hoboken: John Wiley & Sons.
  • Tanveer, A., & Ahmad, S. M. (2023). Design and Testing Of a Compack Inexpensive Prototype Remotely Operated Underwater Vehicle for Shallow Water Operation. Journal of Naval Architecture and Marine Engineering, 20(1), 1-10.
  • Utkin, V. I. (1977). Variable Structure Systems with Sliding Modes. IEEE Transactions On Automatic Control, 22(2), 212-222. https://www.doi.org/10.1109/TAC.1977.1101446
  • Utkin, V. I. (1992). Sliding mode In Control and Optimization. New York, U.S.A.: Springer.
  • Utkin, V. I. (1993). Sliding Mode Control Design Principles and Applications to Electric Drives. IEEE Transactions On Industrial Electronics, 40(1), 23-36. https://www.doi.org/10.1109/41.184818
  • Utkin, V. I., & Parnakh, A. (1978). Sliding Modes and their Application in Variable Structure Systems (Russian) (1st ed.). Mir.
  • Utkin, V. I., & Yang, K. D. (1978). Methods for construction of discontinuity planes in multidimensional variable structure systems. Automat. i Telemekh., (10), 72-77.
  • Utkin, V. I., Guldner, J., & Shi, J. (2009). Sliding Mode Control in Electromechanical Systems (2nd ed.). CRC Press.
  • Wang, B., Liu, C., Chen, S., Dong, S., & Hu, J. (2019). Data-Driven Digital Direct Position Servo Control by Neural Network With Implicit Optimal Control Law Learned From Discrete Optimal Position Tracking Data. IEEE Access, 7, 126962-126972. https://www.doi.org/10.1109/ACCESS.2019.2937993
  • Xiang, Z., & Wei, W. (2021). Design of DC motor position tracking system based on LQR. Journal of Physics: Conference Series, 1887, 012052. https://www.doi.org/10.1088/1742-6596/1887/1/012052
  • Yousef, A. M. (2011). Experimental Set up Verification of Servo Dc Motor Position Control Based On Integral Sliding Mode Control Approach. Journal of Engineering Sciences, 39(5), 1095-1110.
  • Yu, G.-R., Tseng, M.-H., & Lin, Y.-K. (2004, September 2-4). Optimal Positioning Control of a DC Servo Motor Using Sliding Mode. In: Proceedings of the 2004 IEEE International Conference on Control Applications. Taipei. https://www.doi.org/10.1109/CCA.2004.1387223

Year 2023, Volume: 10 Issue: 4, 571 - 592, 31.12.2023

Hakan Kızmaz

https://doi.org/10.54287/gujsa.1393092

Abstract

References

  • Anderson, B. D., & Moore, J. B. (2007). Optimal Control Linear Quadratic Methods (91.12 edition ed.). Dover Publications.
  • Aravind, M. A., Saikumar, N., & Dinesh, N. S. (2017, May 19-21). Optimal position control of a DC motor using LQG with EKF. In: Proceedings of the International Conference on Mechanical, System and Control Engineering (ICMSC), (pp. 149-154). St. Petersburg. https://www.doi.org/10.1109/ICMSC.2017.7959461
  • Burns, R. S. (2001). Advanced Control Engineering. Woburn, England: Butterworth-Heinemann.
  • Davis, J. H. (2002). Luenberger Observers. In: Foundations of Deterministic and Stochastic Control (pp. 245-254). Boston: Birkhäuser. https://www.doi.org/10.1007/978-1-4612-0071-0_8
  • Dorf, R. C., & Bishop, R. H. (2010). Modern Control Systems (12th ed.). (M. J. Horton, A. Gilfillan, A. Dworkin, S. Disanno, G. Dulles, & D. Sandin, Eds.) New Jersey, U.S.A.: Pearson.
  • Dreyfus, S. (1962). Variational problems with inequality constraints. Journal of Mathematical Analysis and Applications, 4(2), 297-308. https://doi.org/10.1016/0022-247X(62)90056-2
  • Durdu, A., & Dursun, E. H. (2019). Sliding Mode Control for Position Tracking of Servo System with a Variable Loaded DC Motor. Elektronika Ir Elekctrotechnika, 25(4), 8-16. https://www.doi.org/10.5755/j01.eie.25.4.23964
  • Edwards, C., & Spurgeon, S. K. (1998). Sliding Mode Control Theory and Applications. Boca Raton: CRC Press.
  • Eli, S. C., Idoniboyeobu, D. C., & Braide, S. L. (2023). Performance Evaluation of D.C. Motor Speed Using Sliding Mode Controller (SMC). Journal of Emerging Trends in Electrical Engineering, 5(3), 1-6.
  • Feng, X., Liu, S., Yuan, Q., Xiao, J., & Zhao, D. (2023). Research on wheel-legged robot based on LQR and ADRC. Scientific Reports, 13(15122). https://www.doi.org/10.1038/s41598-023-41462-1
  • Franklin, G. F., Powell, J. D., & Emami-Naeini, A. (2009). Feedback Control of Dynamic Systems (6th ed.). New Jersey, U.S.A.: Pearson Education.
  • Gao, W., & Hung, J. C. (1993). Variable Structure Control of Nonlinear Systems A New Approach. IEEE Transactions On Industrial Electronics, 40(1), 45-55.
  • Golub, G. H., & Loan, C. F. (2013). Matrix Computations. Baltimore: The Johns Hopkins University Press.
  • Gorczyca, P., Hajduk, K., & Kołek, K. (2011). Optimal Control of A Laboratory DC Servo Motor. Pomiary automatyka Robotyka, 15(12), 129-134.
  • He, S., Duan, X., Qu, X., & Xiao, J. (2023). Kinematic modeling and motion control of a parallel robotic antenna pedestal. Robotica, 41(11), 3275-3295. https://www.doi.org/10.1017/S0263574723000917
  • Jouini, M., Dhahri, S., & Sellami, A. (2019). Combination of integral sliding mode control design with optimal feedback control for nonlinear uncertain systems. Transactions of the Institute of Measurement and Control, 41(5), 1331-1339. https://www.doi.org/10.1177/0142331218777562
  • Khomenko, M., Voytenko, V., & Vagapov, Y. (2013). Neural Network based Optimal Control of Dc Motor Positioning System. International Journal of Automation and Control, 7(1/2), 83-104. https://www.doi.org/10.1504/IJAAC.2013.055097
  • Kirk, D. E. (1998). Optimal Control Theory: An Introduction. New York, Mineola, USA: Dover Publications, Inc.
  • Krause, P. C., Wasynczuk, O., & Sudhoff, S. D. (2002). Analysis of Electric Machinery and Drive Systems (2nd ed.). U.S.A.: John Wiley & Sons.
  • Krishnan, R. (2001). Electric Motor Drives Modeling Analysis and Control (1st ed.). New Jersey: Prentice Hall.
  • Maghfiroh, H., Anwar, M., Anwar, M., & Ma'arif, A. (2022). Improved LQR Control Using PSO Optimization and Kalman Filter Estimator. IEEE Access, 10, 18330-18337. https://www.doi.org/10.1109/ACCESS.2022.3149951
  • Mamta, & Singh, B. (2020). Optimal Control of DC motor using Equilibrium Optimization Algorithm. International Journal of Engineering Research & Technology (IJERT), 9(5), 1272-1275.
  • Mondal, R., & Dey, J. (2020). Performance Analysis and Implementation of Fractional Order 2-DOF Control on Cart-Inverted Pendulum System. IEEE Transactions On Industry Applications, 56(6), 7055-7066.
  • Mukhopadhyay, S. (1978). P.I.D. Equivalent of Optimal Regulator. Elektronics Letters, 14(25), 821-822.
  • Naidu, D. S. (2002). Optimal Control Systems (1st ed.). CRC Press.
  • Nise, N. S. (2011). Control System Engineering (6th ed.). Pomona, U.S.A.: John Wiley & Sons, Inc.
  • Ogata, K. (2010). Modern Control Engineering (5th ed.). Natick: Pearson.
  • Paraskevopoulos, P. N. (2002). Modern Control Engineering. CRC Press.
  • Pontryagin, L. S. (1986). The Mathematical Theory of Optimal Processes (Vol. Vol 4). Switzerland: Gordon and Breach Science Publishers.
  • Pratama, G. N., Setiawan, N., Saputra, S. A., Pambudi, L., Umam, A. D., & Hermansah, M. N. (2022). Optimal Quadratic Regulator PID for Motor DC. Journal of Physics: Conference Series, 2406(012002), 1-7. https://www.doi.org/10.1088/1742-6596/2406/1/012002
  • Rasheed, L. T. (2020). Optimal Tuning of Linear Quadratic Regulator Controller Using Ant Colony Optimization Algorithm for Position Control of a Permanent Magnet Dc Motor. Iraqi Journal of Computers, Communications, Control & Systems Engineering (IJCCCE), 20(3), 29-41. https://www.doi.org/10.33103/uot.ijccce.20.3.3
  • Ruderman, M., Krettek, J., Hoffmann, F., & Bertram, T. (2008). Optimal State Space Control of DC Motor. IFAC Proceedings Volumes, 41(2), 5796-5801. https://www.doi.org/10.3182/20080706-5-KR-1001.00977
  • Saputra, D. D., Ma'arif, A., Maghfiroh, H., Baballe, M. A., Tusset, A. M., Sharkawy, A.-N., & Majdoubi, R. (2023). Performance Evaluation of Sliding Mode Control (SMC) for DC Motor Speed Control. Jurnal Ilmiah Teknik Elektro Komputer dan Informatika (JITEKI), 9(2), 502-510.
  • Simon, D. (2006). Optimal State Estimation. Hoboken: John Wiley & Sons.
  • Tanveer, A., & Ahmad, S. M. (2023). Design and Testing Of a Compack Inexpensive Prototype Remotely Operated Underwater Vehicle for Shallow Water Operation. Journal of Naval Architecture and Marine Engineering, 20(1), 1-10.
  • Utkin, V. I. (1977). Variable Structure Systems with Sliding Modes. IEEE Transactions On Automatic Control, 22(2), 212-222. https://www.doi.org/10.1109/TAC.1977.1101446
  • Utkin, V. I. (1992). Sliding mode In Control and Optimization. New York, U.S.A.: Springer.
  • Utkin, V. I. (1993). Sliding Mode Control Design Principles and Applications to Electric Drives. IEEE Transactions On Industrial Electronics, 40(1), 23-36. https://www.doi.org/10.1109/41.184818
  • Utkin, V. I., & Parnakh, A. (1978). Sliding Modes and their Application in Variable Structure Systems (Russian) (1st ed.). Mir.
  • Utkin, V. I., & Yang, K. D. (1978). Methods for construction of discontinuity planes in multidimensional variable structure systems. Automat. i Telemekh., (10), 72-77.
  • Utkin, V. I., Guldner, J., & Shi, J. (2009). Sliding Mode Control in Electromechanical Systems (2nd ed.). CRC Press.
  • Wang, B., Liu, C., Chen, S., Dong, S., & Hu, J. (2019). Data-Driven Digital Direct Position Servo Control by Neural Network With Implicit Optimal Control Law Learned From Discrete Optimal Position Tracking Data. IEEE Access, 7, 126962-126972. https://www.doi.org/10.1109/ACCESS.2019.2937993
  • Xiang, Z., & Wei, W. (2021). Design of DC motor position tracking system based on LQR. Journal of Physics: Conference Series, 1887, 012052. https://www.doi.org/10.1088/1742-6596/1887/1/012052
  • Yousef, A. M. (2011). Experimental Set up Verification of Servo Dc Motor Position Control Based On Integral Sliding Mode Control Approach. Journal of Engineering Sciences, 39(5), 1095-1110.
  • Yu, G.-R., Tseng, M.-H., & Lin, Y.-K. (2004, September 2-4). Optimal Positioning Control of a DC Servo Motor Using Sliding Mode. In: Proceedings of the 2004 IEEE International Conference on Control Applications. Taipei. https://www.doi.org/10.1109/CCA.2004.1387223
There are 45 citations in total.

Details

Primary Language English
Subjects Control Theoryand Applications
Journal Section Electronics, Sensors and Digital Hardware
Authors

Hakan Kızmaz 0000-0001-7680-7191

Publication Date December 31, 2023
Submission Date November 19, 2023
Acceptance Date December 26, 2023
Published in Issue Year 2023 Volume: 10 Issue: 4

Cite

APA Kızmaz, H. (2023). Comparative Analysis of Optimal Control Strategies: LQR, PID, and Sliding Mode Control for DC Motor Position Performance. Gazi University Journal of Science Part A: Engineering and Innovation, 10(4), 571-592. https://doi.org/10.54287/gujsa.1393092
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